Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space - when the parameters vary. This involves investigating the manifold and, in particular, understanding whether it is close to a low-dimensional affine space. This leads to the notion of Kolmogorov $N$-width that consists of evaluating to which extent the best choice of a vectorial space of dimension $N$ approximates ${\mathcal S}$ well enough. If a good approximation of elements in ${\mathcal S}$ can be done with some well-chosen vectorial space of dimension $N$ -- provided $N$ is not too large -- then a ``reduced'' basis can be proposed that leads to a Galerkin type method for the approximation of any element in ${\mathcal S}$. In many cases, however, the Kolmogorov $N$-width is not so small, even if the parameter set lies in a space of small dimension yielding a manifold with small dimension. In terms of complexity reduction, this gap between the small dimension of the manifold and the large Kolmogorov $N$-width can be explained by the fact that the Kolmogorov $N$-width is linear while, in contrast, the dependency in the parameter is, most often, non-linear. There have been many contributions aiming at reconciling these two statements, either based on deterministic or AI approaches. We investigate here further a new paradigm that, in some sense, merges these two aspects: the nonlinear compressive reduced basisapproximation. We focus on a simple multiparameter problem and illustrate rigorously that the complexity associated with the approximation of the solution to the parameter dependant PDE is directly related to the number of parameters rather than the Kolmogorov $N$-width.

翻译：用于逼近参数依赖偏微分方程（PDE）解的降基方法，其基础在于学习解集的结构——该解集被视为某个函数空间中的流形 ${\mathcal S}$——当参数变化时。这涉及对流形的探究，特别是理解其是否接近一个低维仿射空间。这引出了 Kolmogorov $N$-宽度的概念，即评估在何种程度上，维度为 $N$ 的最佳向量空间选择能够足够好地逼近 ${\mathcal S}$。如果 ${\mathcal S}$ 中的元素可以通过某个精心选择的 $N$ 维向量空间（前提是 $N$ 不太大）得到良好逼近，那么就可以提出一个“降阶”基，从而导出一个用于逼近 ${\mathcal S}$ 中任意元素的 Galerkin 类型方法。然而，在许多情况下，即使参数集位于一个低维空间中，从而产生一个低维流形，Kolmogorov $N$-宽度也并不小。在复杂度降低方面，流形的小维度与大的 Kolmogorov $N$-宽度之间的这种差距可以解释为：Kolmogorov $N$-宽度是线性的，而相比之下，对参数的依赖性通常是非线性的。已有许多研究致力于调和这两种表述，无论是基于确定性方法还是人工智能（AI）方法。本文进一步研究了一种在某种意义上融合了这两个方面的新范式：非线性压缩降基逼近。我们聚焦于一个简单的多参数问题，并严格论证了逼近该参数依赖 PDE 解所需的复杂度，直接与参数数量相关，而非 Kolmogorov $N$-宽度。